The simple truth about photography that nothing is more important then light, or more precisly the quility of it. Sometimes you see amazing photos made with mobile phone, they look gorgeous and you wander how good the camera in your phone is. What they don't tell you though is that yes, they used mobile phone, but they also used couple thousands of lightning equipment.
So let's break down what kind of modifers we have and when do we use them.
(This will be beginner guide and will not delve into specifics, best brands and such. But for those whom only shoot with natural light this could be helpful or give them an idea on what will they need)
Probably the cheapest, easiests and most used piece for many photographers. Gold, silver and white are most used but you can also buy zebra (silver and gold mixed). White is used to reflect light from light source ( we will go with the sun) in a more natural manner. Silver on the other hand is stronger, more constrasty light. Also the downside of silver is blinding a model, she will squint and close her eyes. You could also combine it with a scrim for even more natural, soft light.
When buying they come either as 5 in 1 or 7 in 1 (circle, oval) and they are foldable. Or they are a metal skeleton with fabric applied over it. The price will go from 40 to 200$
A scrim is a device used by photographers, to modify properties of light usually to dampen the intensity of sunlight or direct flash. They come in size ranging tremendously and also cost from 50$ up to 500$. It all depends on the size, quality and brand. Sunbounce is wildly known as the industry standard. You can use Scrim with reflector to diffuse the direct sun and then add fill light with a silver or white reflector. You can get the kit of both for around 150$ with an ok construction and together they do wonders.
Are portable flashes originally made for use on your camera. With many wireless triggers available today that isn't the case anymore. The primary advantage of them is in their weight and compactness. I personally am not a fan of on camera flash so we will be talking about it being off camera. Usually with that you will lose TTL but don't worry, manual mode is a great thing to learn. There are many cheap adapters for combining umbrella or even softbox and a stand with a speedlight and they all work. Sort of.
The main problem of an off camera speedlight is simply a power output. Even the most powerful top ranger models of Nikon and Canon only output around 76-80w of power. A portable off camera flashes are somewhere around 400-600w. So roughly 10 times more powerful. Of course you don't always need this power and if you aren't planning on over powering the sun but rather giving your subject just a subtle fill then it can work. A good simple flash is around 100$, add 10$ adapter, 10$ silver umbrella and 30$ stand and for 150$ you can have a surprisingly ok first off camera light.
A monolight is a self-contained photographic flash lighting unit usually found in a studio. They usually range from 300 all the way to 1200w. That means that even on the low end they are around 4 times more powerful then a Speedlight. But the more important benefit is definitely in the sheer volume of different modifiers you can work with. You can use spot grid and have a narrow beam of direct light, beauty dish for softer but still contrast look, or softbox for diffused more natural light. With over 200 modifiers available you can always get exactly what you want. Cost will go from around 300$ up to 2000$
Off camera flash (battery powered)
Two main difference between monolight and off camera flash are portability and battery. Usually with portable flashes you standalone battery pack you plug into a flash head. Some exceptions like Profoto B1 have battery integrated into the head itself. As such they are standalone flashes with no external plugins like power or trancivers needed. The upside is portability, shooting with them anywhere and having 10 times more power then a speedlight. Downside usually involves 200-300 full power flashes for one battery and cost. They are usually double the cost of a monolight. But when it comes to outside production you really do not have any choice. Some new models even have LED modeling light which could probably be somehow useful for videographers. Cost of one off camera flash is from 900$ all the way to 7000$ for Broncolor mobile 1200. So again, not cheap.